By Samiou E.

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**Additional resources for 2-step nilpotent Lie groups of higher rank**

**Example text**

Then a is b i n o m i a l iff for all n > l . ~> Proof: kt(a) Use the i d e n t i t y = d___log(kt(a)) dt (l+t) a iff the r i g h t h a n d Corollary: For all m E2Z, ~'n(m)=m. side = ~ ( - l ) n ~ n + l ( a ) t n. Then rl--o is ~ ( - l ) n a t n - a l+t I 49 Hence, sub-k-ring will given R 1 of R b y R 1 = at least The any k-ring include Ix I ~n(x)=x, the unit first p r o p o s i t i o n in v e r i f y i n g that v a r i o u s we n e e d a definition: element r£R, R, we can p i c k pre-k-rings include This binomial subring a copy of ~.

W 2 .... )) = Thus (rl,r 2 .... ) where r = n ~dw dh d can identify rI = w 1 2 r2 = w I + 2w 2 3 r3 = wI = r4 + 3w 3 4 wI + 2w22 + 4w 4 etc. If R is t o r s i o n - f r e e , M is a one-one map and we W R with its image Proposition: there M(WR) WR(M) c R . , under G. w i t h sum and product integer coefficients l F n depends [wl I i d i v i d e s on t w o n]) sets of variables: Indeed such j M ( ( W l , w 2 .... ) ) + M ( ( w l , w 2 .... ) )= M ( ( F I ( W l , W l (here in R e. that e )'F2(wI'w2'wI'w~) [wi I i d i v i d e s n] ....

S h o u l d remark, i, b u t U n d e r this d e f i n i t i o n 1 1 2 its s q u a r e A X A = A is not, is m u l t i p l i c a t i v e . powers, seems n e a r l y as p l a u s i b l e . t h a t t h i s c a t e g o r y of v a r i e t i e s is a g o o d e x a m p l e of a c a t e g o r y w h i c h has symmetric but whose Grothendieck and H e n c e R is not a k - r i n g and no o t h e r d e f i ~ i t i o n however, _affine sums, ring products, is not over k and a k-ring. 54 W e n o w u s e the t h e o r e m to c o n s t r u c t k-rings.